Question

Consider the following observations regarding the median of a data set.

Observation I : The median is defined only for categorical data.

Observation II : For an odd number of data points, median is the central value of the ordered data set, whereas for an even number of data points, median is the mean of two central values of the ordered data set.

Observation III : Unlike mode, median may or may not be one of the entries of the data set.

• A. Only observations I and II are correct.
• B. Only observation II is correct.
• C. Only observations II and III are correct.
• D. All the observations are correct.

Answer 1

The correct option is C Only observations II and III are correct.

Let's go through the mentioned observations one by one.

Observation I :

Determination of median of a data set involves arranging the data in an ascending or descending order. Hence, it is only possible for numerical data; not for categorical data.

$\therefore$ Observation I is incorrect.

Observation II :

For a data set with odd number of entries, a single middle value exists for the ordered set. Whereas, for an even numbered ordered data set, two middle values exist.

By definition, for an even numbered ordered data set, median is the mean of two central values.

$\therefore$ Observation II is correct.

Observation III :

For an odd numbered ordered data set, median is the central value. Hence, in this case, median is one of the data points of the data set.

For an even numbered ordered data set, as the median is the mean of two central values, it may or may not be one of the entries of the data set.

$\bullet$ For an example, consider the following ordered data set:

$5$

$\underset{–}{6}$

$\underset{–}{8}$

$10$

Here, the central values are $6$ and $8$.

$\therefore$ Median= Mean of two central values
$\Rightarrow$ Median = $\frac{6+8}{2}=\frac{14}{2}=7$

But the median $7$ is not present in the given data set $\{5,6,8,10\}$.

$\bullet$ Consider, another arranged data set in which two central values are same.

$3$

$\underset{–}{5}$

$\underset{–}{5}$

$8$

Here, the central values are $5$ and $5$.

$\therefore$ Median = $\frac{5+5}{2}=\frac{10}{2}=5$

The median $5$ of this even numbered data set is same as one of the data points.

Hence, the median may or may not be one of the entries of the data set.

$\therefore$ Only observations II and III are correct.